Scaling Behavior of Thermally Driven Fractures in Deep Low‐Permeability Formations: A Plane Strain Model With 1‐D Heat Conduction

Injection of cold fluids through/into deep formations may cause significant cooling, thermal stress, and possible thermal fracturing. In this study, the thermal fracturing of low‐permeability formations under one‐dimensional heat conduction was investigated using a plane strain model. Dimensionless governing equations, with dimensionless fracture length L $\mathcal{L}$ , aperture Ω ${\Omega}$ , spacing D $\mathcal{D}$ , time τ $\tau $ , and effective confining stress T $\mathcal{T}$ , were derived. Solution of single thermal fracture was derived analytically, while solution of multiple fractures with constant (or dynamic) spacing were obtained using the displacement discontinuity method (and stability analysis). For single fracture, L(τ,T) $\mathcal{L}(\tau ,\mathcal{T})$ increases nonlinearly with τ $\sqrt{\tau }$ and then transitions to scaling law L=f(T)τ $\mathcal{L}=f(\mathcal{T})\sqrt{\tau }$ , indicating that late‐time fracture length increases linearly with the square root of cooling time. For constantly spaced fractures, L(τ,T,D) $\mathcal{L}(\tau ,\mathcal{T},\mathcal{D})$ deviates from the single‐fracture solution at a later τ $\tau $ for a larger D $\mathcal{D}$ , showing slower propagation under inter‐fracture stress interaction. For dynamically spaced fractures, fracture arrest induced by stress interaction was determined by the stability analysis; the fully transient solution provides evolution of dimensionless fracture length, spacing, aperture, and pattern; a similar scaling law, L=f′(T)τ $\mathcal{L}={f}^{\prime }(\mathcal{T})\sqrt{\tau }$ with f′(T)